TERM-ORDER

ACL2 must occasionally choose which of two terms is syntactically
smaller. The need for such a choice arises, for example, when using
equality hypotheses in conjectures (the smaller term is substituted
for the larger elsewhere in the formula), in stopping loops in
permutative rewrite rules (see loop-stopper), and in choosing
the order in which to try to cancel the addends in linear arithmetic
inequalities. When this notion of syntactic size is needed, ACL2
uses ``term order.'' Popularly speaking, term order is just a
lexicographic ordering on terms. But the situation is actually more
complicated.

We define term order only with respect to terms in translated form.
See trans. Constants are viewed as built up by pseudo-function
applications, as described at the end of this documentation.

Term1 comes before term2 in the term order iff

(a) the number of variable occurrences in term1 is less than that in
term2, or

(b) the numbers of variable occurrences in the two terms are equal
but the number of function applications in term1 is less than that
in term2, or

(c) the numbers of variable occurrences in the two terms are equal,
the numbers of functions applications in the two terms are equal,
but the number of pseudo-function applications in term1 is less
than that in term2, or

(d) the numbers of variable occurrences in the two terms are equal,
the numbers of functions applications in the two terms are equal,
the numbers of pseudo-function applications in the two terms are equal,
and term1 comes before term2 in a lexicographic ordering, lexorder,
based their structure as Lisp objects: see lexorder.

The function term-order, when applied to the translations of two
ACL2 terms, returns t iff the first is ``less than or equal'' to the
second in the term order.

By ``number of variable occurrences'' we do not mean ``number of
distinct variables'' but ``number of times a variable symbol is
mentioned.'' (cons x x) has two variable occurrences, not one.
Thus, perhaps counterintuitively, a large term that contains only
one variable occurrence, e.g., (standard-char-p (car (reverse x)))
comes before (cons x x) in the term order.

Since constants contain no variable occurrences and non-constant
expressions must contain at least one variable occurrence, constants
come before non-constants in the term order, no matter how large the
constants. For example, the list constant

'(monday tuesday wednesday thursday friday)

comes before x in the term order. Because term order is involved
in the control of permutative rewrite rules and used to shift
smaller terms to the left, a set of permutative rules designed to
allow the permutation of any two tips in a tree representing the
nested application of some function will always move the constants
into the left-most tips. Thus,

(+ x 3 (car (reverse klst)) (dx i j)) ,

which in translated form is

(binary-+ x
(binary-+ '3
(binary-+ (dx i j)
(car (reverse klst))))),

will be permuted under the built-in commutativity rules to

(binary-+ '3
(binary-+ x
(binary-+ (car (reverse klst))
(dx i j))))

or

(+ 3 x (car (reverse klst)) (dx i j)).

Two terms with the same numbers of variable occurrences, function
applications, and pseudo-function applications are ordered by
lexicographic means, based on their structures. See lexorder.
Thus, if two terms (member ...) and (reverse ...) contain the same
numbers of variable occurrences and function applications, then the
member term is first in the term order because member comes before
reverse in the term order (which is here reduced to alphabetic
ordering).

It remains to discuss the notion of pseudo-function applications.

Clearly, two constants are ordered using cases (c) and (d) of term
order, since they each contain 0 variable occurrences and no
function calls. This raises the question ``How many function
applications are in a constant?'' Because we regard the number of
function applications as a more fundamental measure of the size of a
constant than lexicographic considerations, we decided that for the
purposes of term order, constants would be seen as being built by
primitive constructor functions. These constructor functions are
not actually defined in ACL2 but merely imagined for the purposes of
term order. We here use suggestive names for these imagined
functions, ignoring entirely the prior use of these names within
ACL2. The imagined applications of these functions are what we
refer to as pseudo-function applications.

The constant function z constructs 0. Positive integers are
constructed from (z) by the successor function, s. Thus 2 is
(s (s (z))) and contains three function applications. 100
contains one hundred and one applications. Negative integers are
constructed from their positive counterparts by -. Thus, -2
is (- (s (s (z)))) and has four applications. Ratios are
constructed by the dyadic function /. Thus, -1/2 is

(/ (- (s (z))) (s (s (z))))

and contains seven applications. Complex rationals are similarly
constructed from rationals. All character objects are considered
primitive and are constructed by constant functions of the same
name. Thus #\a and #\b both contain one application.
Strings are built from the empty string, (o) by the
``string-cons'' function written cs. Thus "AB" is
(cs (#\a) (cs (#\b) (o))) and contains five applications.
Symbols are obtained from strings by ``packing'' the symbol-name
with the unary function p. Thus 'ab is

(p (cs (#\a) (cs (#\b) (o))))

and has six applications. Note that packages are here ignored and
thus 'acl2::ab and 'my-package::ab each contain just six
applications. Finally, conses are built with cons, as usual.
So '(1 . 2) is (cons '1 '2) and contains six applications,
since '1 contains two and '2 contains three. This, for better
or worse, answers the question ``How many function applications are
in a constant?''